Rota-Baxter operators on Clifford semigroups and the Yang-Baxter equation

Abstract

In this paper, we introduce the theory of Rota-Baxter operators on Clifford semigroups, useful tools for obtaining dual weak braces, i.e., triples (S, +, •) where (S, +) and (S,To each algebraic structure is associated a set-theoretic solution of the Yang-Baxter equation that has a behaviour near to the bijectivity and non-degeneracy. Drawing from the theory of Clifford semigroups, we provide methods for constructing dual weak braces and deepen some structural aspects, including the notion of ideal.

Left cancellative left semi-braces of size $pq$ and $2p^2$ and their nilpotency

“…If (B, +) is a group, then (B, +, •) is a skew left brace, and if in addition (B, +) is abelian then (B, +, •) is a left brace. For further generalizations of left braces, we refer the reader to [6,15,16,17,18,23,24,32] In [35,36] Rump completely classified the left braces for which (B, +) is a cyclic group, while in [37] he developed a machinery to construct finite left braces (B, +, •) having fixed group (B, •). In [39] Smocktunowicz also focused on the group (B, •): indeed, she investigated left braces (B, +, •) for which (B, •) is a nilpotent group.…”